A210013 -id:A210013 - cp's OEIS frontend

A203569 Numbers whose digits are a permutation of [0,...,n] and which contain the product of any two adjacent digits as a substring.

0, 10, 102, 120, 201, 210, 1203, 1302, 2013, 2031, 2103, 2130, 3012, 3021, 3102, 3120, 12034, 12043, 20314, 20413, 21304, 21403, 30214, 30412, 31204, 31402, 34012, 34120, 40213, 40312, 41203, 41302, 43012, 43120, 120345, 120543, 203145, 203154, 204153

Offset: 1

M. F. Hasler, Jan 03 2012

The subsequence A198298 corresponding to n=9 was suggested by E. Angelini (cf. link).
If we consider permutations of [1,...,n], the only solutions are { 1, 12, 21, 213, 312, 3412, 4312, 71532486 }.
There are 285 terms.

			The term 12034 is in the sequence since 1*2=2, 2*0=0, 0*3=0 and 3*4=12 are all substrings of 12034. This is the least nontrivial term in the sense that it contains two adjacent digits > 1, which is the case for all solutions > 42000.

Jason Kimberley, Table of n, a(n) for n = 1..285 (complete sequence)
Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]
E. Angelini, 10 different digits, 9 products, Posting to Seqfan List, Jan 03 2012

Cf. A198298, A203566, A210013-A210020.

n_digit_terms(n)={ my(a=[],p=vector(n,i,10^(n-i))~,t);for(i=(n-1)!,n!-1, is_A203565(t=numtoperm(n,i)%n*p) & a=concat(a,t));vecsort(a)}

A198298 Pandigital numbers (A050278) with each product of adjacent digits visible as a substring of the digits.

Original entry on oeis.org

3205486917, 3207154869, 4063297185, 4063792185, 4230567819, 4230915678, 4297630518, 4297631805, 5042976318, 5063297184, 5079246318, 5093271486, 5094236718, 5148609327, 5180429763, 5180792463, 5180942367, 5184063297, 5420796318

Offset: 1

Eric Angelini and Jason Kimberley, Jan 03 2012

There are 58 terms.

			5x4 ("20") is a substring of 5420976318, as are 4x2 ("8"), 2x0 ("0"), 0x9 ("0"), 9x7 ("63"), 7x6 ("42"), 6x3 ("18"), 3x1 ("3") and 1x8 ("8").
4297631805 is also a member (4*2="8"; 2*9="18"; 9*7="63"; 7*6="42"; 6*3="18"; 3*1="3"; 1*8="8"; 8*0="0"; 0*5="0").

Jason Kimberley, Table of n, a(n) for n = 1..58 (complete sequence)
Eric Angelini, 10 different digits, 9 products
Eric Angelini, 10 different digits, 9 products [Cached copy, with permission]
Eric Angelini, 10 different digits, 9 products, Posting to Seqfan List, Jan 03 2012

Cf. A203569, A203566, A210013-A210020.

from itertools import combinations, permutations
def agen():
    c = 0
    digits = list("0123456789")
    for f in digits[1:]:
        rest = digits[:]
        rest.remove(f)
        for p in permutations(rest):
            t = (f, ) + p
            s = "".join(t)
            if all(str(int(t[i])*int(t[i+1])) in s for i in range(9)):
                yield int(s)
afull = list(agen())
print(afull) # Michael S. Branicky, Oct 03 2024

A203566 Numbers that contain the product of any two adjacent digits as a substring, and have at least one pair of adjacent digits > 1.

Original entry on oeis.org

126, 153, 1025, 1052, 1126, 1153, 1260, 1261, 1262, 1530, 1531, 1535, 2045, 2054, 2126, 2137, 2153, 2173, 2204, 2214, 2306, 2316, 2408, 2418, 2510, 2612, 2714, 2816, 2918, 3056, 3065, 3126, 3153, 3206, 3216, 3309, 3319, 3412, 3515, 3618, 4022, 4058, 4085, 4122, 4126, 4153, 4208, 4218

Offset: 1

M. F. Hasler, Jan 03 2012

Inspired by the problem restricted to pandigital numbers suggested by E. Angelini (cf. link).
Any number having no two adjacent digits larger than 1 is trivially in the sequence A203565, which motivated the present sequence.
In the same way, any number obtained from some a(n) of this sequence by adding any number of digits '0' and '1' on either side is again in this sequence (126 -> 1126, 1260, 1261, ...). This suggests that "primitive" numbers of this kind be defined.

			The number 126 is in the sequence since 1*2=2 and 2*6=12 are both substrings of "126".

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]
E. Angelini, 10 different digits, 9 products, Posting to Seqfan List, Jan 03 2012

Cf. A198298, A203569, A210013-A210020.

has(n,m)={ my(p=10^#Str(m)); until( m>n\=10, n%p==m & return(1))}
is_A203566(n)={ my(d,f=0); n>21 & vecsort(d=eval(Vec(Str(n))))[#d-1]>1 & for( i=2,#d, d[i]<2 & i++ & next; d[i-1]>1 | next; has(n,d[i]*d[i-1]) | return; f=1);f }
for( n=22,9999, is_A203566(n) & print1(n","))

A210020 In base 9, numbers n which have 9 distinct digits, do not start with 0, and have property that the product (written in base 9) of any two adjacent digits is a substring of n.

Original entry on oeis.org

247053168, 264057138, 264075138, 264138057, 264138075, 426057138, 426075138, 426138057, 426138075, 531680247, 531680742, 532407168, 532416807, 570264138, 570426138, 571380264, 571380426, 705324168, 708423516, 716084235, 716805324, 742053168, 750264138, 750426138, 751380264, 751380426, 842071635, 842350716, 842351607

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Paul Davalan.

The analog in base 2 is 10; in base 3, 102,120,201,210.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

A338942 Lexicographically earliest sequence of distinct positive terms such that a(n) is present in 3*a(n+1).

Original entry on oeis.org

1, 4, 8, 6, 2, 7, 9, 3, 10, 34, 78, 26, 42, 14, 38, 46, 82, 94, 98, 66, 22, 74, 58, 86, 62, 54, 18, 60, 20, 40, 80, 160, 534, 178, 594, 198, 660, 220, 734, 578, 526, 842, 614, 538, 846, 282, 940, 980, 1660, 5534, 5178, 1726, 5754, 1918, 6394, 8798, 6266, 5422, 8474, 6158, 5386, 8462, 6154, 8718, 2906, 4302, 1434, 478

Offset: 1

Eric Angelini and Carole Dubois, Nov 17 2020

The lexicographically earliest sequence of positive terms such that a(n) is present in 2*a(n+1) is A000351 (the powers of 5).

Conjecture: For n >= 222, a(n) = 100*a(n-16). - Jason Yuen, Dec 22 2024

			a(1) = 1 is present (as a substring) in 12 [= 3 * a(n+1) = 3 * 4];
a(2) = 4 is present in 24 (= 3 * 8);
a(3) = 8 is present in 18 (= 3 * 6);
a(4) = 6 is present in 6 (= 3 * 2); etc.

Carole Dubois, Table of n, a(n) for n = 1..270

Cf. A210013, A000351.

a:=[1]; f:=func; for n in [2..70] do k:=2; while k in a or not f(a[n-1],3*k) do k:=k+1; end while; Append(~a,k); end for; a; // Marius A. Burtea, Nov 19 2020

A210014 Pandigital numbers (A050278) with each product of four adjacent digits visible as a substring of the digits.

Original entry on oeis.org

4327059168, 4613590728, 4613590872, 7860241359, 7860291354, 8490536127, 8760241359, 8760291354

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Marc Falcoz.

			Example for 4327059168: 4*3*2*7= 168; 5*9*1*6=270; 9*1*6*8=432 are substrings.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

A210015 In base 4, numbers n which have 4 distinct digits, do not start with 0, and have property that the product (written in base 4) of any two adjacent digits is a substring of n.

Original entry on oeis.org

1203, 1230, 1302, 2013, 2031, 2103, 2130, 3012, 3021, 3102, 3120

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Paul Davalan.

The analog in base 2 is 10; in base 3, 102,120,201,210.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

A210016 In base 5, numbers n which have 5 distinct digits, do not start with 0, and have property that the product (written in base 5) of any two adjacent digits is a substring of n.

Original entry on oeis.org

13024, 13042, 20314, 20413, 21304, 21403, 24013, 24130, 30214, 30412, 31204, 31402, 40213, 40312, 41203, 41302, 42013, 42130

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Paul Davalan.

The analog in base 2 is 10; in base 3, 102,120,201,210.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

A210017 In base 6, numbers n which have 6 distinct digits, do not start with 0, and have property that the product (written in base 6) of any two adjacent digits is a substring of n.

Original entry on oeis.org

203415, 204315, 205134, 205143, 235104, 250314, 251403, 302514, 305124, 305214, 312405, 314025, 314052, 321045, 321054, 341205, 410235, 431205, 451032, 503124, 512034, 512043, 512403, 520143, 520314, 521403, 541032

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Paul Davalan.

The analog in base 2 is 10; in base 3, 102,120,201,210.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

A210018 In base 7, numbers n which have 7 distinct digits, do not start with 0, and have property that the product (written in base 7) of any two adjacent digits is a substring of n.

Original entry on oeis.org

1504326, 1506234, 1540326, 1543026, 1543260, 2153406, 2340615, 2341506, 2601543, 2603154, 2603415, 2604315, 2615034, 2615043, 2615403, 2615430, 3026154, 3154026, 3260154, 3260415, 3261504, 3261540, 3402615, 3406215, 3415026, 3415062, 4032615, 4053216, 4061325, 4062153, 4062315, 4132506, 4150326, 4150623, 4302615, 4306215, 4315026, 4315062, 4320615, 4321506, 4326015, 4326150, 5321406, 5321604, 6021534, 6023415, 6041325, 6043215, 6053214, 6132504, 6150234, 6150432, 6203415, 6204315, 6215034, 6215043, 6215304, 6215340, 6230415, 6231504, 6234015, 6234150

Offset: 1

N. J. A. Sloane, Mar 16 2012

Computed by Jean-Paul Davalan.

The analog in base 2 is 10; in base 3, 102,120,201,210.

Eric Angelini, 10 different digits, 9 products
E. Angelini, 10 different digits, 9 products [Cached copy, with permission]

A generalization of A198298. Cf. A210013-A210020, A203569, A203566.

cp's OEIS Frontend

A203569 Numbers whose digits are a permutation of [0,...,n] and which contain the product of any two adjacent digits as a substring.

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PARI

A198298 Pandigital numbers (A050278) with each product of adjacent digits visible as a substring of the digits.

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Python

A203566 Numbers that contain the product of any two adjacent digits as a substring, and have at least one pair of adjacent digits > 1.

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PARI

A210020 In base 9, numbers n which have 9 distinct digits, do not start with 0, and have property that the product (written in base 9) of any two adjacent digits is a substring of n.

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A338942 Lexicographically earliest sequence of distinct positive terms such that a(n) is present in 3*a(n+1).

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Magma

A210014 Pandigital numbers (A050278) with each product of four adjacent digits visible as a substring of the digits.

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A210015 In base 4, numbers n which have 4 distinct digits, do not start with 0, and have property that the product (written in base 4) of any two adjacent digits is a substring of n.

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A210016 In base 5, numbers n which have 5 distinct digits, do not start with 0, and have property that the product (written in base 5) of any two adjacent digits is a substring of n.

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A210017 In base 6, numbers n which have 6 distinct digits, do not start with 0, and have property that the product (written in base 6) of any two adjacent digits is a substring of n.

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A210018 In base 7, numbers n which have 7 distinct digits, do not start with 0, and have property that the product (written in base 7) of any two adjacent digits is a substring of n.

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